limit rules of functions

Theorem 1.

Let $f$ and $g$ be two real (http://planetmath.org/RealFunction) or complex functions.  Suppose that there exist the limits  ${lim}_{x\to x_0}f(x)$  and  ${lim}_{x\to x_0}g(x)$.  Then there exist the limits  ${lim}_{x\to x_0}[f(x)\pm g(x)]$,  ${lim}_{x\to x_0}f(x)g(x)$  and, if  ${lim}_{x\to x_0}g(x)\ne 0$,  also  ${lim}_{x\to x_0}f(x)/g(x)$, and

1. ${lim}_{x\to x_0}[f(x)\pm g(x)]={lim}_{x\to x_0}f(x)\pm {lim}_{x\to x_0}g(x),$

2. ${lim}_{x\to x_0}f(x)g(x)={lim}_{x\to x_0}f(x)\cdot {lim}_{x\to x_0}g(x),$

3. ${lim}_{x\to x_0}\frac{f(x)}{g(x)}=\frac{{lim}_{x\to x_0}f(x)}{{lim}_{x\to x_0}g(x)},$

4. ${lim}_{x\to x_0}c=c\mathit{\hspace{1em}}\mathrm{where}c\mathrm{is}\mathrm{a}\mathrm{constant}.$

These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.).

In 1, the domains of $f$ and $g$ could be any topological space (not necessarily $ℝ$ or $ℂ$).

There are limit rules of sequences (http://planetmath.org/Sequence).

As well, one often needs the

Theorem 2.

If there exists the limit  ${lim}_{x\to x_0}f(x)=a$  and if $g$ is continuous at the point  $x=a$, then there exists the limit  ${lim}_{x\to x_0}g(f(x))$, and

$$\underset{x\to x_0}{lim}g(f(x))=g(\underset{x\to x_0}{lim}f(x)).$$